# Stochastic Rating of Storage Systems in Isolated Networks with Increasing Wave Energy Penetration

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## Abstract

**:**

## 1. Introduction

## 2. Test Case Presentation

_{d}represents the power produced by the diesel generator, p

_{farm}the power produced by the wave farm, p

_{L}is the total power consumed by the local loads and p

_{dump}is the power dissipated by the dump load.

_{ES}represents the power exchanged by the ESS at time t, which is assumed to be positive during charge intervals and negative during discharge ones.

#### 2.1. Input Meteorological Data

_{s}, average period of zero up-crossing T

_{z}, and wave direction θ, which were provided with a time resolution of 3 hours. Data of the year 2012 were considered for the present study, to derive the expected wave power production, as explained in detail in Section 4. To match the per-hour resolution of the analysis, the meteorological data were up-sampled by spline interpolation.

#### 2.2. Input Electric Data

## 3. Model of the Diesel Power Generation

_{dmin}= 0.3 of the rated capacity), in order to maintain a suitable efficiency and avoid lifetime reduction and possible fire hazards [17]. However, modern diesel generators utilizing electronic fuel injections to maintain suitably high engine temperature can work relatively well below the above mentioned load limit. For the sake of simplicity, in this paper it is considered that the rated capacity of the diesel generation plant equals the yearly peak load and that the diesel generator is always switched on and operates according to the minimum load constraint. This can be mathematically expressed by writing the power produced by the diesel generator in each time instant, t, as:

## 4. Model of the Wave Power Generation

#### 4.1. Simulation Model of Lifesaver

_{e,i}is the time dependent force due to incident waves, and M denotes the mass of the system:

_{D,i}accounts for the sum of all the damping forces in Equation (6). Here, F

_{r,i}accounts for the time dependent forces on the floater due to radiation of waves. The term F

_{d,I}accounts for non-linear damping terms, mainly the drag forces. ζ

_{i}(t) is the time dependent motion of the floater, C

_{i}is the restoring force matrix accounting for the hydrostatic pressure acting on the floater, and F

_{PTO}(t) is the time dependent force applied from the PTO. The PTO is modeled as a rope and winch system that is tightly moored to the sea floor.

_{s}= 2.75 m / T

_{z}= 6.5 s.

#### 4.2. Simulation for the Array

_{WEC}absorbers placed in a string can thus be described as the sum of the contributions given in Equation (7).

_{WEC}absorbers on a string is the sum of the excitation potential due to incident waves ϕ

_{0}, the diffraction potential due to the interaction of the incident potential with all absorbers at rest ϕ

_{D}, and the radiation potential ϕ

_{R}due to the independent motion of every absorber in every mode of motion with no incident waves present.

_{WEC}+ 1 independent problems to solve. Further the radiation potential from each absorber is separated in six independent modes of motion. The total potential ${\phi}_{{N}_{WEC}}^{i}$ acting on absorber N

_{WEC}in mode i of motion is thus the sum of every other absorbers' radiation and diffraction potential in addition to the diffraction and radiation potential from absorber N

_{WEC}acting on itself in mode i of motion. Combining the six modes of motions for each absorber and allowing all absorbers to interact results in a total of N

_{WEC}x6 independent linear equations to be solved for each wave frequency.

_{WEC}= 7) as a function of array angle in Figure 6. By using the given wave direction as input, the expected array power output can then be found by simply multiplying the power output from each WEC with the corresponding correction factor.

## 5. Wave Energy Capability to Match Electric Demand

_{farm}is the rated power of the entire wave farm and P

_{L}is the maximum power consumption (load) of La Palma. Once the considered power load is defined from real data, the wave power penetration depends on the number of arrays used for wave power production and is then constant for the considered wave farm. In the following analysis two different cases are considered:

- Case 1: Limited wave power penetration (r
_{P}= 0.5); - Case 2: High wave power penetration (r
_{P}= 1).

**Figure 7.**(

**a**) Daily wave energy penetration for considered Case 2; (

**b**) Daily cross-correlation for both considered Case 2 and Case 1.

## 6. Methodology for Energy Storage Sizing

#### 6.1. Data Preprocessing

_{e}and c, offers similar level of accuracy.

_{e}and c values calculated from all the data (Figure 7) can be divided into a corresponding number of divisions, as shown in Figure 8 for the case of N = 49. Then, the actual wave generated power and power consumption profiles, having desired r

_{e}and c must be generated for each scenario. The starting point is considered to be the daily consumption profile with the highest power value in the year and the corresponding wave power profile is generated so that it meets the above requirements. It is worth noting that wave power profiles generated in this way are not unique.

**Figure 8.**(

**a**) Discrete probability distribution of daily wave energy penetration, r

_{e}, for Case 2 (N = 49 scenarios); (

**b**) Discrete probability distribution of daily cross-correlation between wave generated power and power consumption, c, for Case 2 and Case 1 (N = 49 scenarios).

#### 6.2. Storage Sizing Procedure

_{ES}, and power rating, P

_{ES}, of the ESS based on the daily generation and consumption power profiles of the N scenarios, weighted with their occurrence probability.

_{e,fix}and fixed power cost, π

_{p,fix}, associated to the acquisition of the storage device itself.

_{ES,p}and π

_{ES,e}, by applying the basic formula presented here for the power rating:

_{pj}represents the project period in years, r represents the discount rate and the interest is compounded annually. A similar formula is used for the energy rating.

_{ES}, P

_{ES}and E

_{0}, where E

_{0}represents the initial energy condition of the storage system. They are common to all the scenarios since the storage rating must be defined before selecting and deploying the device. First stage variables can be represented in vector form as:

**p**

_{d},

**p**

_{dump},

**p**

_{ES}and are defined for each time interval of the considered scenario z. This can be written as:

_{z}calculates the expected operating cost of power generation over the random variable z and over all the time points t. It is worth underlining that the actual cost of the power produced by the wave farm does not affect the result of the minimization process: it is included in the cost function just to obtain as a result a representative cost of the produced energy.

- Power balance, which corresponds to Equation (2) to be rewritten in vector form in the random variable z;
- Diesel operation limit, corresponding to Equation (3) to be rewritten in vector form in the random variable z;
- Dumping load constraint, corresponding to Equation (4) to be rewritten in vector form in the random variable z.

- 4.
- ESS power rating, which can be expressed as:$$\left|{p}_{ES}(z)\right|\le {P}_{ES}$$
- 5.
- ESS system operation, which can be expressed as:$${e}_{ES,t}(z)={\displaystyle \sum _{j=1}^{t}[{E}_{0}+\gamma *{p}_{ES,j}(z)]}$$$$\gamma =\{\begin{array}{c}\eta \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}if\text{\hspace{0.17em}}{p}_{ES,j}\ge 0\\ 1/\eta \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}if\text{\hspace{0.17em}}{p}_{ES,j}<0\end{array}$$
- 6.
- ESS energy rating, which can be expressed as:$$0\le {e}_{ES}(z)\le {E}_{ES}$$

_{T}, during the following day, the additional constraint on the final energy state is added as follows:

- 7.
- Final ESS energy state, which can be expressed as:$${e}_{T}(z)={E}_{0}$$

## 7. Simulation Results for the Considered Test Case

#### 7.1. Technical-Economic Parameters

_{efix}and π

_{pfix}presented in Table 1 refer to the best case values for Lead-Acid battery technology according to [24]. The corresponding storage efficiency has been set to 0.8.

Type of parameters | N_{pj} (years) | r (%) | π_{d} ($/MWh) | π_{farm} ($/MWh) | P_{dmin} | π_{efix} ($/MWh) | π_{pfix} ($/MW) | η |
---|---|---|---|---|---|---|---|---|

Technical parameters | - | - | - | - | 0.3 | - | - | 0.8 |

Economic parameters | 20 | 8.5 | 600 | 258 | - | 200 | 300 | - |

#### 7.2. Simulation Results

#### 7.2.1. Reference Case

^{®}code for the implementation of above presented optimization process has been written and run. Corresponding results are reported in Table 2 for two different levels of wave power penetration. It can be observed that in case of limited wave power penetration (Case 1) introduction of the ESS is not justified by any reduction in the cost of served energy. This is because, despite the constraint on minimum diesel generated power, the limited contribution of wave power to the total generation limits the amount of energy dissipated in the dump load. In turn, the overall cost of served energy is, as expected, intermediate between the cost of diesel and of wave, since wave energy contribution reduces the energy required to the diesel unit, thus lowering the final price. When wave power penetration increases (Case 2) an excess of generation compared to the consumption is experienced more frequently. Being the load consumption the same as in case 1, the cost of energy when no storage is included is increased, due to the cost of the higher amount of energy wasted in the dump load for the low power cross-correlation. As a consequence of this, the introduction of an energy storage system is advisable and would reduce the cost of served energy by 0.32%.

**Table 2.**Results of the energy storage optimization considering the standard operation strategy for traditional diesel generators (P

_{dmin}= 0.3) for different levels of wave power penetration.

Considered case | E_{ES} optimum (MWh) | P_{ES} optimum (MW) | Cost of energy with optimum storage ($/MWh) | Cost of energy without storage ($/MWh) |
---|---|---|---|---|

Case 1: r_{p} = 0.5 | - | - | - | 577.8 |

Case 2: r_{p} = 1 | 33.9 | 7.8 | 577.5 | 579.4 |

#### 7.2.2. Effect of Diesel Operation Strategy

_{dmin}= 0.2. Corresponding results are reported in Table 3.

**Table 3.**Results of the energy storage optimization considering the advanced operation strategy for diesel generators (P

_{dmin}= 0.2) for different levels of wave power penetration.

Considered case | E_{ES} optimum (MWh) | P_{ES} optimum (MW) | Cost of energy with optimum storage ($/MWh) | Cost of energy without storage ($/MWh) |
---|---|---|---|---|

Case A: r_{p} = 0.5 | - | - | - | 570.8 |

Case B: r_{p} = 1 | 25.5 | 6.1 | 560.1 | 560.9 |

_{dmin}= 0.3.

#### 7.2.3. Effect of Generation Cost Variation

**Figure 9.**Reduction in the cost of energy served for the optimal storage solution in the reference Case 2, for increasing values of the generation costs.

#### 7.2.4. Effect of ESS Cost Variation

_{e,fix and}π

_{p,fix}, corresponding to the state of the art of different storage technologies and to verify, through the above presented optimization process, if they could be a viable solution for the considered wave/diesel test-case.

_{e,fix}

_{and}π

_{p,fix}) are selected according to [24] and they represent the most optimistic data for each technology. All the simulations have been performed using the same value for the ESS efficiency (η = 0.8). Results of such analysis are reported in Figure 10. Red crossed points represent situations where, based on the technical-economic parameters, ESS storage introduction is not convenient or not feasible.

**Figure 10.**ESS technology applicability and optimal sizing based on economic parameters from [24].

_{p,fix}, results in higher values of the optimum power rating (B), which is in line with the current applications of this technology. Economic parameters potentially corresponding to both polysulfide bromide (PSB) batteries and zinc-bromine (Zn-Br) batteries (C) provide an optimal ESS rating that is achievable with PSB, but not with Zn-Br. Finally the yellow marked solution that has been found for vanadium redox (VR) batteries appears to be at the very limit of applicability of such technology, but it could become feasible with next advances in the state of the applied art.

**Figure 11.**ESS technology applicability based on the state of the applied art (reproduced from [25]).

## 8. Discussion and Conclusions

- Energy storage becomes more and more convenient with the increase of the wave power penetration level in the local power balance;
- The less flexible the operation strategy of the main diesel generators is, the more convenient the ESS deployment becomes;
- For the considered reference case, the introduction of the optimally sized ESS could lead to a 0.32% reduction on the cost of served energy. Such percentage can increase up to 1.1% if generation costs (from both diesel and wave) increase by 70%, while storage costs are kept constant.

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Tedeschi, E.; Sjolte, J.; Molinas, M.; Santos, M.
Stochastic Rating of Storage Systems in Isolated Networks with Increasing Wave Energy Penetration. *Energies* **2013**, *6*, 2481-2500.
https://doi.org/10.3390/en6052481

**AMA Style**

Tedeschi E, Sjolte J, Molinas M, Santos M.
Stochastic Rating of Storage Systems in Isolated Networks with Increasing Wave Energy Penetration. *Energies*. 2013; 6(5):2481-2500.
https://doi.org/10.3390/en6052481

**Chicago/Turabian Style**

Tedeschi, Elisabetta, Jonas Sjolte, Marta Molinas, and Maider Santos.
2013. "Stochastic Rating of Storage Systems in Isolated Networks with Increasing Wave Energy Penetration" *Energies* 6, no. 5: 2481-2500.
https://doi.org/10.3390/en6052481